Related papers: On Primes Represented by Quadratic Polynomials

We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…

We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we…

We introduce a new conjecture on products of two distinct primes that would provide a partial answer to a conjecture of McIntosh. Also, $\binom{2p-1}{p-1}-1$ is written in terms of a polynomial in prime $p$ over the integers and we discuss…

This is an exposition of recent developments in the theory of bounded differences between primes. Readers are expected to be beginners of analytic number theory. The present text is a substantially improved and augmented version of the one…

We study an LCM-based analogue of Rowland's GCD-based prime-generating recurrence, introduced by the author in 2008. The multiplicative increments of this sequence are conjectured always to be $1$ or prime, but a complete proof requires a…

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who…

We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest…

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…

By involving some exponential sums related to $\Lambda(n)$ in arithmetic progression, we can obtain some new results for von Mangoldt function over {\bf nonhomogeneous} Beatty sequences in arithmetic progressions, which improve some recent…

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem,…

We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version…

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of…

We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many…

A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the…

In this work, we offer a historical stroll through the vast topic of binary quadratic forms. We begin with a quick review of their history and then an overview of contemporary algebraic developments on the subject.

This paper describes some of the ideas used in the development of our work on small gaps between primes.

In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.

Let $1<c<d$ be two relatively prime integers and $g_{c,d}=cd-c-d$. We confirm, by employing the Hardy--Littlewood method, a 2020 conjecture of Ram\'{\i}rez Alfons\'{\i}n and Ska{\l}ba which states that $$#\left\{p\le g_{c,d}:p\in…

We provide a framework for using elliptic curves with complex multiplication to determine the primality or compositeness of integers that lie in special sequences, in deterministic quasi-quadratic time. We use this to find large primes,…